2 1 Introduction The 3x+1 Problem is perhaps today s most enigmatic unsolved mathematical problem: it can be explained to child who has learned how to divide by and multiply by 3, yet there are relatively few strong results toward solving it. Paul Erdös was correct when he stated, Mathematics is not ready for such problems. The problem is also referred to as the 3n + 1 problem and is associated with the names of Collatz, Hasse, Kakutani, Ulam, Syracuse, and Thwaites. It may be stated in a variety of ways. Defining the Collatz function as 3x + 1 x 1 (mod ) C(x) = x x 0 (mod ), the conjecture states that for each m Z +, there is a k Z + such that C (k) (m) = 1, that is, any positive integer will eventually iterate to 1. Note that an odd number m iterates to 3m + 1 which then iterates to (3m + 1)/. One may therefore compress the dynamics by considering the map 3x+1 x 1 (mod ) T (x) = x x 0 (mod ). The map T is usually favored in the literature. To a much lesser extent some authors work with the most dynamically streamlined 3x + 1 function, F : Z + odd Z+, defined by odd F (x) = 3x + 1 m(3x+1) where m(x) equals the number of factors of contained in 3x+1. While working with F allows one to work only on the odd positive integers, the variability of m seems to prohibit any substantial analysis. This survey reflects the author s view of how work on this problem can be structured. I owe a huge debt to Jeff Lagarias and Günther Wirsching for the important work they have done in bringing this problem forward. paper of Lagarias[45](1985) thoroughly catalogued earlier results, made copious The

3 3 connections, and developed many new lines of attack; it has justly become the classical reference for this problem. Wirsching s book [87](1998) begins with a strong survey, followed by several chapters of his own noteworthy analysis. Lagarias has also maintained an annotated bibliography [48](1998) of work on this problem, another valuable resource. This current survey would have been much more difficult to write in the absence of these significant contributions. This survey is not meant to be exhaustive, but rather is complementary to the work of Lagarias and Wirsching. Where I believed there was significant new work in a given area, I included earlier contributions for the sake of completeness. Some areas which have not seen recent development, such as the interesting work connected to functional equations, cellular automata, and the origin of the problem, have not been mentioned; the reader may consult Wirsching s book [87](1998). Numerical Investigations and Stopping Time The structure of the positive integers forces any orbit of T to iterate to one of the following: 1. the trivial cycle {1, }. a non-trivial cycle 3. infinity (the orbit is divergent) The 3x + 1 Problem claims that option 1 occurs in all cases. Oliveira e Silva[61, 6](1999,000) proved that this holds for all numbers n < This was accomplished with two 133MHz and two 66MHz DEC Alpha computers and using 14.4 CPU years. This computation ended in April 000. Roosendaal[65](003) claims to have improved this to n = His calculations continue, with the aid of many others in this distributed-computer project. The record for proving the non-existence of non-trivial cycles is that any such cycle must have length no less than 7,500,658. This was derived with

4 4 the help of numerical results like those in the last paragraph coupled with the theory of continued fractions see a section 5 for more on cycles. There is a natural algorithm for checking that all numbers up to some N iterate to one. First, check that every positive integer up to N 1 iterates to one, then consider the iterates of N. Once the iterates go below N, you are done. For this reason, one considers the so-called stopping time of n, that is, the number of steps needed to iterate below n : σ(n) = inf{k : T (k) (n) < n}. Related to this is the total stopping time, the number of steps needed to iterate to 1: σ (n) = inf{k : T (k) (n) = 1}. One considers the height of n, namely, the highest point to which n iterates: h(n) = sup{t (k) (n) : k Z + }. Note that if n is in a divergent trajectory, then σ (n) = h(n) =. These functions may be surprisingly large even for small values of n. For example, σ(7) = 59, σ (7) = 111, h(7) = 93. The orbit of 7 is depicted in Figure 1. Roosendaal[65](003) has computed various records for these functions 1. Various results about consecutive numbers with the same height are catalogued by Wirsching[87, pp.1 ](1998)..1 Stopping Time The natural algorithm mentioned earlier can be rephrased: the 3x + 1 problem is true if and only if every positive integer has a finite stopping time. Terras[77, 1 Roosdendaal has different names for these functions, and he applies them to the map C(x).

5 Figure 1: Orbit starting at 7.

6 6 78](1976,1979) has proven that the set of positive integers with finite stopping time has density one. Specifically, he showed that the limit 1 F (k) := lim {n m : σ(n) k} m m exists for each k Z + and lim k F (k) = 1. A shorter proof was provided by Everett[31](1977). Lagarias[45](1985) proved a result regarding the speed of convergence: F (k) 1 ηk for all k Z +, where η = 1 H(θ), H(x) = x log(x) (1 x) log(1 x) and θ = (log 3) 1. He uses this to constrain any possible divergent orbits: {n Z + : n x, σ(n) = } c1 x 1 η for some positive constant c 1. This implies that any divergent trajectory cannot diverge too slowly. Along these lines, Garcia and Tal[33](1999) have recently proven that the density of any divergent orbit is zero. This result holds for more general sets and more general maps. Along different lines, Venturini[81](1989) showed that for every ρ > 0, the set {n Z + : T (k) (n) < ρn for some k} has density one. Allouche[1](1979) showed that {n Z + : T (k) (n) < n c for some k} has density one for c > 3/ log This was improved by Korec[39](1994) who obtained the same result with c > log Not surprisingly, results of a probabilistic nature abound concerning these 3x + 1 functions. An assumption often made is that after many iterations, the next iterate has an equal chance of being either even or odd. Wagon[84](1985) argues that the average stopping time for odd n (with the function C(x)) approaches a constant, specifically, the number [1 + i + i log 3] i=1 c i [i log 3] where c i is the number of sequences containing 3/ and 1/ with exactly i 3/ s such that the product of the whole sequence is less than 1, but the product

7 7 of any initial sequence is greater than 1. Note that this formula bypasses the pesky +1, perhaps justified asymptotically. This seems to be borne out in Wagon s numerical testing of stopping times for odd numbers up to n = 10 9, which matches well with the approximation given above.. Total Stopping Time Results regarding the total stopping time are also plenteous. Applegate and Lagarias[9](00) make use of two new auxiliary functions: the stopping time ratio γ(n) := σ (n) log n, and the ones-ratio ρ(n) for convergent sequences, defined as the ratio of the number of odd terms in the first σ (n) iterates divided by σ (n). It is easy to see that γ(n) 1/ log for all n, with equality only when n = k. Stronger inequalities are γ(n) 1 log ρ(n) log 3 for any convergent trajectory, while if ρ(n) 0.61, then for any positive ɛ, γ(n) 1 log ρ(n) log 3 + ɛ for all sufficiently large n. If one assumes that the ones-ratio equals 1/ (which is equivalent to the equal probability of encountering an even or an odd after many iterates), we have that the average value of γ(n) is / log(4/3) 6.951, as has been observed by Shanks[69](1965), Crandall[6](1978), Lagarias[45](1985), Rawsthorne[64](1985), Lagarias and Weiss[47](199), and Borovkov and Pfeifer[15](000). Experimental evidence supports this observation. Compare this with the upper bounds suggested by the stochastic models of Lagarias and Weiss[47](199): lim sup γ(n) n Applegate and Lagarias[9](00) note from records of Roosendaal[65](003) what is apparently the largest known value of γ: n = 7, 19, 136, 416, 377, 36, 71, 195

8 8 produces σ (n) = 1848 and γ(n) Applegate and Lagarias seek a lower bound for γ which holds infinitely often. Using the well-known fact that T (k) ( k 1) = 3 k 1 (see, for example, Kuttler[44](1994)), they note that γ( k 1) log + log 3 (log ) They go on to show that there are infinitely many converging n whose ones-ratio is at least 14/9, hence giving the lower bound γ(n) 9 9 log 14 log for infinitely many n. Though the proof involves an extensive computational search on the Collatz tree to depth 60, the authors note with surprise that the probabilistic average of γ(n) approximately was not attained. Roosendaal[65](003) defines a function which he calls the residue of a number x Z +, denoted Res(x). Suppose x is a convergent C-trajectory with E even terms and O odd terms before reaching one. Then the residue is defined as Res(x) := E x3 O. Roosendall notes that Res(993) = , and that this is the highest residue attained for all x < 3. He conjectures that this holds for all x Z +. Zarnowski[89](001) recasts the 3x + 1 problem as one with Markov chains. Let g be the slightly altered function g(x) = 1 x = 1, 3x+1 x 1 (mod ), x > 1 x x 0 (mod ) e n the column vector whose n th entry is one and all other entries zero, and the transition matrix P be defined by 1 if g(i) = j, P ij = 0 otherwise. This gives a correspondence between g (k) (n) and e T n P k, and the 3x+1 Problem is true if lim P k = [1 0 0 ], k

9 Figure : The Collatz graph. where 1 and 0 represent constant column vectors. Zarnowski goes on to show that the structure of a certain generalized inverse X of I P encodes the total stopping times of Z +. For pictorial representations of stopping times for an extension of T, see Dumont and Reiter[9]..3 The Collatz Graph and Predecessor Sets The topic of stopping times is intricately linked with the Collatz tree or Collatz graph, the directed graph whose vertices are predecessors of one via the map T. It is depicted in Figure.3. The structure of the Collatz graph has attracted some attention. Andaloro[4](00) studied results about the connectedness of subsets of the Collatz graph. Urvoy[79](000)

10 10 proved that the Collatz graph is non-regular, in the sense that it does not have a decidable monadic second order theory; this is related to the work of Conway mentioned in Section 4. Instead of analyzing how fast iterates approach one, one may consider the set of numbers which approach a given number a, that is, the predecessor set of a: P T (a) := {b Z + : T (k) (b) = a for some k Z + } Since such sets are obviously infinitely large, one may measure those terms in the predecessor set not exceeding a given bound x, that is, Z a (x) := {n P T (a) : n x}. The size of Z a (x) was first studied by Crandall[6](1979), who proved the existence of some c > 0 such that Z 1 (x) > x c, x sufficiently large. Wirsching[87, p.4](1998) notes that this result extends to Z a (x) for all a 0 mod 3. Using the tree-search method of Crandall, Sander[67](1990) gave a specific lower bound, c = 0.5, and Applegate and Lagarias[6](1995) extended this to c = Using functional difference inequalities, Krasikov[41](1989) introduced a different approach and obtained c = 3/ Wirsching[85](1993) used the same approach to obtain c = Applegate and Lagarias[7](1995) superseded these results with Z 1 (x) > x 0.81, for sufficiently large x, by enhancing Krasikov s approach with nonlinear programming. Recently, Krasikov and Lagarias[4](00) streamlined this approach to obtain Z 1 (x) > x 0.84, x sufficiently large. Wirsching[87](1998) has pushed these types of results in a different direction. On a seemingly different course, for j, k Z + let R j,k denote the set of all sums of the form α0 + α1 3 + α αj 3 j

13 13 to obtain some distributional information regarding trajectories of F. Let 1 m M, x 0 and ( m ) ω = log(f (m) (x 0 )) log(x 0 ) + m( log log 3). M M If 0 t 1, then ω(t) behaves like a Wiener trajectory. Related to these results is the connection made in Blecksmith et al.[13](1998) between the 3x+1 problem and 3-smooth representations of numbers. A number n Z + has a 3-smooth representation if and only if there exist integers {a i } and {b i } such that n = k ai 3 bi, a 1 > a > > a k 0, 0 b 1 < b < < b k. i=1 Note the similarity with the terms considered by Wirsching in the last section. These numbers were studied at least as early as Ramanujan. A 3-smooth representation of n is special of level k if n = 3 k + 3 k 1 a a k 1 + a k in which every power of 3 appears up to 3 k. For a fixed k, each n has at most one such representation see Lagarias[46](1990), who credits the proof to Don Coppersmith. Blecksmith et al. then offer the reader to prove that m Z + iterates to 1 under C if and only if there are integers e and f such that the positive integer n = e 3 f m has a special 3-smooth representation of level k = f 1. The choice of e and f is not unique, if it exists. 4 Reduction to Residue Classes and Other Sets Once one is convinced that the 3x + 1 problem is true, a natural approach is to find subsets S Z + such that proving the conjecture on S implies it is true on Z +. It is obvious this holds if S = {x : x 3 mod 4} since numbers in the other residue classes decrease after one or two iterations. Puddu[63](1986) and Cadogan[19](1984) showed that this works if S = {x : x 1 mod 4}. The work of Böhm and Sontacchi[14](1978) implies that m odd T -iterates of x

14 Figure 3: Dynamics of Integers (a) mod 3 and (b) mod 4. equals (3/) m (x + 1) 1, hence odds must eventually become even. Coupling this with the dynamics of integers in Z 4 (see Figure 3b) implies both S = {x : x 1 mod 4} and S = {x : x mod 4} are sufficient. Similarly, Figure 3a indicates S = {x : x 1 mod 3} or S = {x : x mod 3} suffice. Andaloro[3](000) has improved these to S = {x : x 1 mod 16}. All of these sets have an easily computed positive density, the lowest being Andaloro s set S at 1/16. Korec and Znam[40](1987) significantly improve this by showing the sufficiency of the set S = {x : x a mod p n } where p is an odd prime, a is a primitive root mod p, p a, and n Z +. This set has density p n which can be made arbitrarily small. In a similar vein, Yang[88](1998) proved the sufficiency of the set {n : n } (4k 1) mod k+ for any fixed k Z +. Korec and Znam also claim to have a sufficient set with density zero, but details were not provided. To this end, a recent result of Monks[57](00) is noteworthy. The last section showed how difficult it is to find a usable closed form expression for T (k) (x). If the +1 was omitted from the iterations, this would yield T (k) (x) = 3 m x/ k, where m is the number of odd terms in the first k iterations of x. Monks[57](00) has proven that there are infinitely many linear versions of the 3x + 1 problem. An example given

15 15 is the map R(n) = 1 11n if 11 n n if 15 n and NOT A 5 17n if 17 n and NOT A 4 5n if 5 n and NOT A 6 1n if 1 n and NOT A 7 13n if 13 n and NOT A 1 7n if 7 n and NOT A 33 4 n if 4 n and NOT A 5 n if n and NOT A 7n otherwise where NOT A means none of the above conditions hold. Monks shows the 3x+1 problem is true if and only if for every positive integer n the R-orbit of n contains. The proof uses Conway s Fractran language[5](1987) which Conway used [4](197) to prove the existence of a similar map whose long-term behavior on the integers was algorithmically undecidable. Connecting this material back to sufficient sets, one notes that the density of the set { n : n Z + } is zero (albeit, one has a much more complicated map). 5 Cycles Studying the structure of any possible cycles of T has received much attention. Letting Ω be a cycle of T, and Ω odd, Ω even denote the odd and even terms in Ω, one may rearrange the equation x = T (x) x Ω x Ω to obtain x = x Ω even x Ω odd x + Ω odd. This was noted by Chamberland[1](1999) and Monks[57](00). Using modular arithmetic, Figure 3 indicates the dynamics of integers under T both in Z 3 and Z 4. Since no cycles may have all of its elements of the form

16 16 0 mod 3, 0 mod 4, or 3 mod 4, the figures imply that no integer cycles (except {0}) have elements of the form form 0 mod 3. Also, the number of terms in a cycle congruent to 1 mod 4 equals the number congruent to mod 4. This analysis was presented by Chamberland[1](1999). An early cycles result concerns a special class of T -cycles called circuits. A circuit is a cycle which may be written as k odd elements followed by l even elements. Davison[7](1976) showed that there is a one-to-one correspondence between circuits and solutions (k, l, h) in positive integers such that ( k+l 3 k )h = l 1. (1) It was later shown using continued fractions and transcendental number theory (see Steiner[71](1977), Rozier[66](1990)) that equation (1) has only the solution (1, 1, 1). This implies that {1, } is the only circuit. For general cycles of T, Böhm and Sontacchi[14](1978) showed that x Z + is in an n-cycle of T if and only if there are integers 0 v 0 v 1 v m = n such that x = m 1 1 n 3 m 3 m k v k k=0 A similar result is derived by B. Seifert[68](1988). Eliahou[30](1993) has given some strong results concerning any non-trivial cycle Ω of T. Letting Ω 0 denote the odd terms in Ω, he showed ( log ) Ω ( M Ω o log ) m where m and M are the smallest and largest terms in Ω. Note that for large cycles this implies Ω Ω 0 log 3. Eliahou uses this in conjunction with the Diophantine approximation of log 3 and the numerical bound m > 40 to show that Ω = a b c where a, b, c are nonnegative integers, b 1 and ac = 0. Similar results were found by Chisala[](1994) and Halbeisen and Hungerbühler[35](1997). This ()

17 17 approach was pushed the farthest by Tempkin and Arteaga[75](1997). They tighten the relations in () and use a better lower bound on m to obtain Ω = a b c where a, b, c are nonnegative integers, b 1 and ac = 0. It is not apparent how the even-odd dissection of a cycle used in these results may be extended to a finer dissection of the terms in a cycle, say, mod 4. Related to this, Brox[17](000) has proven that there are finitely many cycles such that σ 1 < log(σ 1 + σ 3 ) where σ i equals the number of terms in a cycle congruent to i mod 4. 6 Extending T to Larger Spaces A common problem-solving technique is to imbed a problem into a larger class of problems and use techniques appropriate to that new space. Much work has been done along these lines for the 3x + 1 problem. The subsequent subsections detail work done in increasingly larger spaces. 6.1 The Integers Z The first natural extension of T is to all of Z. The definition of T suffices to cover this case. One soon finds three new cycles: {0}, { 5, 7, 10}, and the long cycle { 17, 5, 37, 55, 8, 41, 61, 91, 136, 68, 34}. These new cycles could also be obtained (without minus signs) if one considered the map define T (n) = T ( n), which corresponds to the 3x 1 problem. It is conjectured that these cycles are all the cycles of T on Z. This problem was considered by Seifert[68](1988).

18 18 6. Rational Numbers with Odd Denominators Just as the study of the 3x-1 problem of the last subsection is equivalent to the 3x + 1 problem on Z, Lagarias[46](1990) has extended the 3x + 1 to the rationals by considering the class of maps T k (x) = positive k ±1 mod 6 with (x, k) = 1. 3x+k x 1 (mod ) x x 0 (mod ) He has shown that cycles for T k correspond to rational cycles x/k for the 3x + 1 function T. Lagarias proves there are integer cycles of T k for infinitely many k (with estimates on the number of cycles and bounds on their lengths) and conjectures that there are integer cycles for all k. Halbeisen and Hungerbühler[35](1997) derived similar bounds as Eliahou[30](1993) on cycle lengths for rational cycles. 6.3 The Ring of -adic Integers Z The next extension is to the ring Z of -adic integers consisting of infinite binary sequences of the form a = a 0 + a 1 + a + = a j j where a j {0, 1} for all non-negative integers j. Congruence is defined by a a 0 mod. Chisala[](1994) extends C to Z but then restricts his attention to the rationals. He derives interesting restrictions on rational cycles, for example, if j=0 m is the least element of a positive rational cycle, then m > m log 3 m 3. A more developed extension of T, initiated by Lagarias[45](1985), defines a/ if a 0 mod T : Z Z, T (a) := (3a + 1)/ if a 1 mod

20 0 Monks and Yazinski also extend the results of Eliahou[30](1993) and Lagarias[45](1985) concerning the density of odd points in an orbit. Let κ n (x) denote the number of ones in the first n-digits of the parity vector x. If x Q odd eventually enters an n-periodic orbit, then ln() ln(3 + 1/m) lim κ n (x) ln() n n ln(3 + 1/M) where m, M are the least and greatest cyclic elements in the eventual cycle. If x Q odd diverges, then ln() κ n (x) lim inf ln(3) n n. Monks and Yazinski define another extension of T, namely ξ : Z Z as Ω(x) x 1 (mod ) ξ(x) = x x 0 (mod ) and prove that the 3x + 1 problem is equivalent to having the number one in the ξ-orbit of every positive. 6.4 The Gaussian Integers and Z [i] Joseph[38](1998) extends T further to Z [i], defining T as α/, if α [0] (3α + 1)/, if α [1] T (α) = (3α + i)/, if α [i] (3α i)/, if α [1 + i] where [x] denotes the equivalence class of x in Z [i]/z [i]. Joseph shows that T is not conjugate to T T via a Z -module isomorphism, but is topologically conjugate to T T. Arguing akin to results of the last subsection, Joseph shows that T is chaotic (in the sense of Devaney). Kucinski[43](000) studies cycles of Joseph s extension T restricted to the Gaussian integers Z[i]. 6.5 The Real Line lr A further extension of T to the real line lr is interesting in that it allows tools from the study of iterating continuous maps. In an unpublished paper,

21 1 Tempkin[76](1993) studies what is geometrically the simplest such extension: the straight-line extension to the Collatz function C: C(x), x Z L(x) = C( x ) + (x x )(C( x ) C( x )), x Z (5n )x + n(10n 3), x [n 1, n] =. (5n + 4)x n(10n + 7), x [n, n + 1] Tempkin proves that on each interval [n, n+1], n Z +, L has periodic points of every possible period. Capitalizing on the fact that iterates of piecewise linear functions are piecewise linear, he also shows that every eventually periodic point of L is rational, every rational is either eventually periodic or divergent, and rationals of the form k/5 with k 0 mod 5 are divergent. Tempkin also mentions a smooth extension to C, namely E(x) := 7x x + 4 but conducts no specific analysis with it. similar extension to T : f(x) := x ( πx ) cos + 3x + 1 = x x cos(π(x + 1)) Chamberland[0](1996) studies a cos(πx). ( sin πx ) Chamberland shows that any cycle on Z + must be locally attractive. By also showing that the Schwarzian derivative of f is negative on lr +, this implies the long-term dynamics of almost all points coincides with the long-term dynamics of the critical points. One quickly finds that there are two attracting cycles, A 1 := {1, }, A := { , }. Chamberland conjectures that these are the only two attractive cycles of f on lr +. This is equivalent to the 3x + 1 problem. It is also shown that a monitonically increasing divergent orbit exists. Chamberland compactifies the map via the homeomorphism σ(x) = 1/x on [µ 1, ) (µ 1 is the first positive

22 Figure 4: Julia set of Chamberland s map extended to the complex plane fixed point of the map f), yielding a dynamically equivalent map h defined on [0, µ 1 ] as h(x) = 4x 4+x (+x) cos(πx), x (0, µ 1] 0, x = 0. Lastly, Chamberland makes statements regarding any general extension of f: it must have 3-cycle, a homoclinic orbit (snap-back repeller), and a monotonically increasing divergent trajectory. To illustrate how chaos figures into this extension, Ken Monks replaced x with z in Chamberland s map and numerically generated the filled-in Julia set. A portion of the set is indicated in Figure 4. In a more recent paper, Dumont and Reiter[8](003) produce similar results for the real extension f(x) := 1 ( ) 3 sin (πx/) x + sin (πx/). The authors have produced several other extensions, as well as figures representing stopping times; see Dumont and Reiter[9](001). Borovkov and Pfeifer[15](000) also show that any continuous extension of T has periodic orbits of every period by arguing that the map is turbulent: there exist compact intervals A 1 and A such that A 1 A = f(a 1 ) f(a ).

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EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime

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Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

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Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

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I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

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Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

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1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

David Harvey (joint work with Edgar Costa, NYU) University of New South Wales 25th October 2011 The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers

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